HMMT 2024 U18

An ordered pair $(a, b)$ of positive integers is called spicy if $\operatorname{gcd}(a+b, a b+1)=1$. Compute the probability that both $(99, n)$ and $(101, n)$ are spicy when $n$ is chosen from $\{1,2, \ldots, 2024$ ! $\}$ uniformly at random.